Massive mediator boson, a game changer, so is the weak force fundamentally different from the other forces? I have read this question:
How it is possible to mediate a weak interactions by very heavy particle such as Boson? 
How can a pion have a mass, given it's a "field mediator" and created/destroyed continuously? 
Can virtual particle have mass? 
Can/do Virtual Particles have mass/energy? 
Why don't virtual particles violate conservation of mass/energy? 
Now EM forces are mediated by virtual photons.
Gravity is mediated by theoretical virtual gravitons.
The strong force is mediated by virtual gluons.
Now all these are massless bosons.
Yes I understand that virtual particles are just a mathematical formula to describe the effects of the force field.
But the weak force is different. It is mediated by massive particels, W and Z bosons.
And yes I understand that the massiveness of these bosons cause the short range of the weak force.
What I do not understand is how can virtual particles be massive and still mediate the force? It is not a real particle. How can something not real have rest mass while mediating an interaction? Virtual particles are just a mathematical model, how can they have rest mass? How do you experimentally prove that a virtual particle has rest mass?
Question:


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*Does the massiveness of the W and Z bozons mean that the weak force is fundamentally different from the other forces? How can virtual particles be massive while mediating an interaction? 

*How do we experimentally prove that virtual W and Z bozons have rest mass?
 A: 
Does the massiveness of the W and Z bozons mean that the weak force is fundamentally different from the other forces? How can virtual particles be massive while mediating an interaction?

As you have stated, virtual means mathematical tool. The four vector of a virtual particle changes according to the kinematics of the incoming and outgoing particles, mass can even be negative. What makes the difference at our time in the cosmos is that symmetry breaking has given a physical mass to free Z and W. This mass enters in the propagator representing the virtual particle in the Feynman diagram . 
In both position space and momentum space propagators , the mass of the virtual particle enters in the denominator. When the virtual exchange has a small mass the total probability contributed to the interaction can be large, photon, electron (not to forget there are a lot of virtual particles in feynman diagrams).

The variables are under the integral of the Feynman diagram. This is to show that the real mass of the virtual particle exchanged in the diagram enters in the denominator, thus when large, diminishing the probability of the interaction the Feynman diagram represents.
With the enormous mass of the Z and W the diagram is suppressed, which is what makes  the weak interaction , low probability of the final integral calculation.  Before symmetry breaking, all forces were equal in this sense as the masses are zero. 

How do we experimentally prove that virtual W and Z bozons have rest mass?

Virtual are not experimentally measurable except by the calculation of the total probability for the experiment fitting the data, i.e. indirectly. The model of feynman diagrams for the calculations has been validated by all experiments up to now, and if we find a violation, we look for extensions of the standard model and not for a change in mathematics.
A: The weak force is inherently different from the other forces. While the other three are transmitted by massless mediators, the weak force is indeed mediated by the massive $W$'s and $Z^0$.
This is a conspicuous difference. Remember though that once the strong force was thought to be mediated by massive pions. Then it was realized that the "old" strong force was a residue force of the new strong force (governed by QCD).
I think this is also the case with the nuclear force. It's a residue force (like the old strong force) of a more fundamental force (like the new strong force) mediated by massless particles.
This new (hypothetical though) force is called the hypercolor force which finds an incarnation in the Rishon Model, a preon model introduced by Haim Harari in the eighties. 
In Harari's model all interactions are mediated by massless particles. You can compare the old strong force (conveyed by massive particles) with the current weak force (also conveyed by massive particles) and the new strong force (mediated by massless gluons) with the (hypothetically) new force of hyper-colored rishons (also mediated by massless particle: the hyper color gluons).
So, in this case, the symmetry between the four forces is restored. Opponents will say that once (when the temperature was high enough) there was a symmetry between the forces (all forces were equal, although this can until now only be said, in the context of a GUT of the three basic forces except for gravity), which was spontaneously broken, resulting in four (three) forces of which one (i.e. the weak force) had massive mediators.
The theory of the electroweak unification is rather messy though, so I'm inclined to say that the weak force is inherently different than the other three (two). 
